WASHINGTON — Senior Republican Party leaders began urging their most imperiled incumbents on Wednesday to speak out about the wrongdoing surrounding President Trump, with Representative Tom Cole, a former House Republican campaign chairman, warning, “Where there’s smoke, and there’s a lot of smoke, there may well be fire.”

Democrats face their own pressure to shed their cautious midterm strategy and hammer the opposition for fostering what Democratic leaders are labeling “a culture of corruption” that starts at Mr. Trump and cascades through two indicted House Republicans to a series of smaller scandals breaking out in the party’s backbenches.

By urging some candidates to speak out or at least stay silent, Republican leaders who gravely fear losing control of the House risked opening the first significant rift between the Trump White House and the Republican-controlled Capitol.

“Anybody who says this is not disturbing is not being honest,” said Mr. Cole, an Oklahoman and the former head of the National Republican Congressional Committee, adding, “so my advice to any candidate would be: Keep your powder dry and don’t rush to attack or defend anybody because you just don’t know enough to have a reaction that you can still defend three months from now.”

On cue, Representative Carlos Curbelo, Republican of Florida, called the accumulation of Trump-related scandals a “sad chapter in our country’s politics” and said that “no one is above the law.” Mr. Curbelo, in a tough re-election fight, also reproached the president for his caustic attacks on the special counsel, Robert S. Mueller III.

“He’s making a major mistake by attacking the Mueller probe in such a personal way,” said Mr. Curbelo, who represents a district that Hillary Clinton won handily in 2016. “The best thing for everyone, especially if the White House is so confident that the president will be absolved in this process, is to let the process continue.”

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Representative Jerrold Nadler of New York, the top Democrat on the House Judiciary Committee, in June in Washington. “This is getting deeper and deeper, and it’s going to get more and more serious,” he said.CreditJ. Scott Applewhite/Associated Press

Representative John Katko, running a difficult race in central New York, told The Post-Standard in Syracuse on Wednesday that mounting legal trouble around the president “raises it to another level of concern, there’s no question about it.”

“Wherever the facts go, if they go towards the president or someone in the White House, they’re not above the law,” Mr. Katko said, voicing support for the Mueller inquiry. “No one is.”

Nearly as alarming to House Republicans were the mushrooming scandals in their own chamber, misdeeds that may dampen turnout among conservative voters.

But the summer eruption of apparent Republican malfeasance has some in the party arguing that Democrats should make corruption more central.

“There’s no way this won’t matter in a whole bunch of races out there, and Democrats need to be talking about this everywhere,” said Senator Christopher S. Murphy of Connecticut, invoking the 2006 campaign, when “late-breaking corruption scandals, on top of an unpopular president, tipped the House and the Senate.”

Or, as Mayor Rahm Emanuel of Chicago, a former congressman and the architect of the 2006 House Democratic takeover, put it: “Trump is the issue, and rather than draining the swamp, he brought it with him.”

Democrats moved on Wednesday to elevate the theme. Representative Nancy Pelosi of California, the House Democratic leader, distributed a public letter to her colleagues arguing that the Trump-era capital had “become a cesspool of self-enrichment, secret money and ethical blindness” and that House Republicans were offering only a “blind eye to the corruption and criminality at the heart of President Trump’s inner circle.”

But in the same letter, Ms. Pelosi also said that Democrats “must also stay focused on delivering our strong economic message.”

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Senator Lindsey Graham, Republican of South Carolina, on Tuesday in Washington. “Campaign finance violations — I don’t know what will come from that, but the thing that will hurt the president the most is if, in fact, his campaign did coordinate with a foreign government like Russia,” he said.CreditErin Schaff for The New York Times

Democratic strategists overseeing the midterm elections said they would use corruption against those Republicans tarred by any wrongdoing, but they were not yet convinced that focusing on what some in the electorate view as unending Washington drama would be more effective than highlighting policy differences.

“For a lot of voters, the Donald Trump circus is not enough to get them to vote Democratic in the midterms,” said Alixandria Lapp, who runs the main House Democratic “super PAC.”

Senator Sherrod Brown, a Democrat running for re-election in Ohio, which Mr. Trump won handily in 2016, put it this way: “Voters see this, and the people who don’t like Trump know it, and the people who do like Trump know it. I think my job is to make the case that the Republicans have cut taxes for the rich. They are going to raise the eligible age for Medicare. And they try to take health care away from people and Democrats stand on the other side.”

Democrats also worry that employing a Trump-tinged message about corruption will only prompt more questions about whether they would use a new House majority to impeach Mr. Trump, a campaign that could rile an otherwise demoralized Republican base. Democratic leaders have studiously avoided the “I”-word for months amid liberal outcry, preferring to shift the burden onto Republicans who have all but ceased conducting oversight of the Trump administration.

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Since the news broke of Michael Cohen’s guilty plea, we received lots of questions from our readers. Adam Liptak, the Times’s Supreme Court reporter, answers them.CreditCreditDoug Mills/The New York Times

But the criminal implications of Mr. Cohen’s guilty plea will raise the pressure on Democrats from their own base.

“We need to see what Mueller comes up with,” Representative Jerrold Nadler of New York, the top Democrat on the House Judiciary Committee, said about the potential for an impeachment inquiry, allowing that “this is getting deeper and deeper, and it’s going to get more and more serious.”

As news of a Cohen plea deal was circulating Tuesday, House Democratic leaders urged members on a private call to stay on message, avoiding the topic of impeachment. Instead, Democrats will cast themselves as offering a check and balance on the president, a message that their polling indicates voters respond to favorably.

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Senator John Cornyn of Texas, the No. 2 Senate Republican, on Tuesday at the Capitol. He repeatedly stepped around questions from reporters about the implications of Mr. Cohen’s pleas.CreditErin Schaff for The New York Times

But this is an election year in which the battles for the House and Senate are being fought on drastically different terrain, with many of the most competitive House contests taking place in suburbs that revile Mr. Trump and control of the Senate hinging on a group of Democrats from more rural states that he won overwhelmingly.

So even as House Republicans suggested that their most at-risk lawmakers should tread gingerly around the embattled president, some of their Senate counterparts taunted Democrats for trying to curry favor with Mr. Trump.

“You’ve got a number of red-state Democrats that are trying to get closer to the president,” said Senator Cory Gardner of Colorado, the chairman of the Senate Republican campaign arm, who flew with Mr. Trump to his rally in West Virginia in the hours of Tuesday’s convictions. “They are trying to run as close to the president as they can.”

Some Democrats from pro-Trump states said they had no immediate plans to highlight the matter of the president authorizing an undisclosed payoff to a pornographic film actress with whom he had an affair.

“If asked, I absolutely will talk about it,” said Senator Jon Tester of Montana, adding that it is not currently a front and center campaign issue. “I think people are still trying to sort through it.”

Given the health of the economy and relative stability abroad, Mr. Trump’s approval ratings are already significantly lower than what virtually any other president would be enjoying at this moment of his administration. And if there is one predictable element of this otherwise unpredictable presidency, it is that some new story will detonate in the days or weeks ahead, pushing the last eruption off the home page and television chyron.

Indeed, Republicans are wishing that the Manafort and Cohen felonies only wind up being this week’s version of the Omarosa tapes — the latest ephemeral drama to captivate the capital but barely faze voters. But party officials are apprehensive about what more Mr. Cohen may reveal on his way to a likely prison sentence.

Even more worrisome to Republicans is what damage Mr. Trump may do to himself. If the president is seen as thwarting Mr. Mueller’s investigation, either by terminating the special counsel or pardoning allies who are implicated, it would create a far more serious upheaval and force Republican lawmakers into a confrontation they have long avoided.

“Of course you can imagine the president doing things that would be counterproductive,” said Senator Bill Cassidy, Republican of Louisiana, before adding with evident hope in his voice, “but you also could imagine the president saying something but never acting on it, just venting.”